Dealing with insurance after not-at-fault crash—OEM parts + injury settlement advice needed

I was driving normally when another driver entered my path without stopping. I hit the brakes immediately, but I was already too close, so my car skidded and we collided. The front right side of my car took the damage (bumper, fender, headlamp, rim, and possibly more—they’re still inspecting it).

The other driver admitted fault at the scene, and their insurance is AAA. They’ve already reported the accident. When I spoke with the insurance company, I explained what happened and mentioned that I’ve had back pain since the crash.

They asked where I wanted the car repaired, and I chose Caliber Collision. After the estimate, the shop told me the insurance company wants to use aftermarket and recycled parts. I asked them to request OEM (genuine Toyota) parts instead, but the adjuster called me and said they won’t approve OEM unless aftermarket parts aren’t available—it’s their policy. I haven’t agreed to repairs yet because of this.

A few more details:

\- I was driving my cousin’s car (it’s under his name and policy, and I’m not listed), but since the other driver is 100% at fault, their insurance is accepting liability.

\- I’ve been dealing with back pain since the accident and have been seeing a chiropractor. I do have bills and notes from treatment.

My questions:

1. Is there any way to push back and get OEM parts approved instead of aftermarket/recycled?

2. Should I hold off on repairs until this is resolved?

3. Regarding my injury, is it reasonable to try to settle with the adjuster for around $3,500 for medical/pain and suffering, given I have chiropractor documentation?

4. Anything else I should be doing right now to protect myself in this situation?

Appreciate any advice from people who’ve dealt with similar situations.